Matrix Addition: Find A+D

In this guide, we'll walk through how to add two matrices, $A$ and $D$, together. Matrix addition is a fundamental operation in linear algebra, and it's quite straightforward as long as the matrices have the same dimensions. Let's dive in!

Understanding Matrix Addition

Before we jump into the specific problem, let's quickly recap the rules for matrix addition. To add two matrices, they must have the same number of rows and columns. If matrix $A$ has dimensions $m \times n$ and matrix $D$ also has dimensions $m \times n$, then we can add them. The resulting matrix, let's call it $A+D$, will also have dimensions $m \times n$. To find each element in $A+D$, we simply add the corresponding elements in $A$ and $D$.

Mathematically, if $A = [a_{ij}]$ and $D = [d_{ij}]$, then $A+D = [a_{ij} + d_{ij}]$, where $i$ represents the row number and $j$ represents the column number.

Detailed Explanation with Examples

Let's consider two matrices, $A$ and $B$, of the same dimensions, say $2 \times 2$:

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

To find $A + B$, we add the corresponding elements:

$A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

The element in the first row and first column of $A+B$ is the sum of the elements in the first row and first column of $A$ and $B$ (i.e., $1 + 5 = 6$). Similarly, for the other elements.

Now, let's consider another example with $3 \times 2$ matrices:

$A = \begin{bmatrix} 9 & 10 \\ 11 & 12 \\ 13 & 14 \end{bmatrix}$ and $B = \begin{bmatrix} 15 & 16 \\ 17 & 18 \\ 19 & 20 \end{bmatrix}$

Then, $A + B$ is:

$A + B = \begin{bmatrix} 9+15 & 10+16 \\ 11+17 & 12+18 \\ 13+19 & 14+20 \end{bmatrix} = \begin{bmatrix} 24 & 26 \\ 28 & 30 \\ 32 & 34 \end{bmatrix}$

Make sure that the matrices have the same dimensions, or you cannot perform the addition. If you try to add matrices of different sizes, it's like trying to add apples and oranges – it just doesn't work in the context of matrix operations!

Problem: Finding A+D

We are given two matrices:

$A=\begin{bmatrix} 5 & -1 \\ 4 & -2 \\ 6 & 5 \end{bmatrix}$ and $D=\begin{bmatrix} 2 & 0 \\ -6 & 5 \\ 3 & 3 \end{bmatrix}$

Both matrices are $3 \times 2$, meaning they have 3 rows and 2 columns. Therefore, we can add them together.

To find $A+D$, we add the corresponding elements in each matrix:

$A+D = \begin{bmatrix} 5+2 & -1+0 \\ 4+(-6) & -2+5 \\ 6+3 & 5+3 \end{bmatrix}$

Now, let's perform the additions:

$A+D = \begin{bmatrix} 7 & -1 \\ -2 & 3 \\ 9 & 8 \end{bmatrix}$

So, the sum of the matrices $A$ and $D$ is:

$\begin{bmatrix} 7 & -1 \\ -2 & 3 \\ 9 & 8 \end{bmatrix}$

Step-by-Step Solution

1. Check Dimensions: Verify that both matrices have the same dimensions. In this case, both $A$ and $D$ are $3 \times 2$ matrices.
2. Add Corresponding Elements: Add the elements in the same positions in the two matrices.
   • Top-left element: $5 + 2 = 7$
   • Top-right element: $-1 + 0 = -1$
   • Middle-left element: $4 + (-6) = -2$
   • Middle-right element: $-2 + 5 = 3$
   • Bottom-left element: $6 + 3 = 9$
   • Bottom-right element: $5 + 3 = 8$
3. Form the Resultant Matrix: Combine the results to form the new matrix.

$A+D = \begin{bmatrix} 7 & -1 \\ -2 & 3 \\ 9 & 8 \end{bmatrix}$

Common Mistakes to Avoid

• Adding Matrices with Different Dimensions: Always ensure that the matrices have the same dimensions before attempting to add them. If they don't, the addition is undefined.
• Incorrectly Adding Elements: Double-check that you are adding the corresponding elements. It's easy to make a mistake if you're not careful, especially with larger matrices.
• Sign Errors: Pay close attention to the signs of the numbers. Adding a negative number is the same as subtracting, so be mindful of that!

Practice Problems

To solidify your understanding, here are a few practice problems:

1. Let $P = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $Q = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$. Find $P+Q$.
2. Let $R = \begin{bmatrix} -1 & 0 \\ 2 & -3 \\ 4 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 5 & -2 \\ -3 & 0 \\ 1 & -1 \end{bmatrix}$. Find $R+S$.
3. Let $X = \begin{bmatrix} 10 & -5 \\ 2 & 8 \end{bmatrix}$ and $Y = \begin{bmatrix} -3 & 6 \\ 9 & -1 \end{bmatrix}$. Find $X+Y$.

Conclusion

Matrix addition is a fundamental operation in linear algebra. By ensuring that the matrices have the same dimensions and adding their corresponding elements, you can easily find the sum of two matrices. Keep practicing, and you'll become a matrix addition master in no time! Remember always to double-check your work to avoid common mistakes, like adding matrices of different dimensions or making sign errors. So go ahead, give those practice problems a shot, and reinforce your understanding of matrix addition. You've got this!